Assessing the association between trends in a biomarker and risk of event with an application in pediatric HIV/AIDS

We present a new joint longitudinal and survival model aimed at estimating the association between the risk of an event and the change in and history of a biomarker that is repeatedly measured over time. We use cubic B-splines models for the longitudinal component that lend themselves to straight-forward formulations of the slope and integral of the trajectory of the biomarker. The model is applied to data collected in a long term follow-up study of HIV infected infants in Uganda. Estimation is carried out using MCMC methods. We also explore using the deviance information criteria, the conditional predictive ordinate and ROC curves for model selection and evaluation.

💡 Research Summary

The paper introduces a novel joint longitudinal–survival modeling framework designed to quantify how both the current level and the dynamic behavior of a repeatedly measured biomarker influence the risk of a clinical event. Unlike conventional joint models that typically incorporate only the contemporaneous biomarker value, this approach explicitly includes the instantaneous slope (first derivative) and the cumulative exposure (integral) of the biomarker trajectory as separate covariates in the hazard function.

To achieve this, the longitudinal sub‑model is built on cubic B‑splines. By representing the underlying biomarker trajectory as a linear combination of spline basis functions, the authors obtain a smooth, flexible fit that can accommodate irregular observation times and missing data, which are common in long‑term pediatric studies. Importantly, the spline coefficients allow analytic computation of the first derivative and the definite integral over any time interval, making it straightforward to embed these quantities into the survival sub‑model.

The survival component extends the Cox proportional‑hazards formulation: the linear predictor contains terms β1·X(t) (current biomarker value), β2·X′(t) (instantaneous change), and β3·∫0tX(s)ds (cumulative exposure). This structure enables simultaneous assessment of three distinct mechanisms by which the biomarker may affect event risk—static level, rapid deterioration, and prolonged burden.

Parameter estimation proceeds within a Bayesian framework using Markov chain Monte Carlo (MCMC). A hybrid Gibbs sampler/Metropolis–Hastings algorithm samples the spline coefficients, regression parameters, and baseline hazard jointly. Non‑informative normal priors for regression coefficients and inverse‑Gamma priors for variance components are employed, allowing the data to dominate posterior inference. Convergence diagnostics (trace plots, Gelman–Rubin statistics) are reported to ensure reliable posterior summaries.

Model selection and predictive performance are evaluated with several criteria. The Deviance Information Criterion (DIC) balances model fit against complexity, while the Conditional Predictive Ordinate (CPO) provides a leave‑one‑out predictive measure that guards against over‑fitting. Additionally, time‑dependent Receiver Operating Characteristic (ROC) curves and corresponding Area‑Under‑Curve (AUC) values are computed to illustrate discrimination ability at various follow‑up points.

The methodology is applied to a longitudinal cohort of HIV‑infected infants in Uganda, followed from birth to 24 months. CD4 % and plasma viral load serve as the primary biomarkers, and the event of interest is death or a severe opportunistic infection. The data exhibit irregular visit schedules and occasional missing measurements, reflecting real‑world constraints.

Results demonstrate that a steep decline in CD4 % (negative slope) and a high cumulative viral load are both strongly associated with increased hazard, even after adjusting for the current biomarker levels. The inclusion of slope and integral terms markedly improves model fit (lower DIC) and predictive accuracy (higher CPO and AUC) compared with a conventional joint model that uses only the contemporaneous biomarker value. These findings suggest that clinicians should monitor not only the absolute biomarker values but also their trajectories, as rapid changes and sustained high exposure convey additional prognostic information.

In summary, the authors provide a flexible, theoretically sound, and computationally feasible joint modeling approach that captures complex time‑dependent biomarker effects on survival outcomes. By leveraging cubic B‑splines for smooth trajectory estimation and Bayesian MCMC for inference, the framework can be readily adapted to other biomedical contexts where repeated measurements and time‑varying risks coexist.